= Optimal turn count for fixed winding volume
{title2=$R_{\rm coil}=R_s$}
For fixed mean turn length and fixed conductor volume, a coil of $N$ turns has conductor length proportional to $N$ and cross-sectional area proportional to $1/N$. Its <electrical resistance> is therefore $R_{\rm coil}=aN^2$. With source <electromotive force> $\mathcal E$ and <internal resistance> $R_s$, suppose the small-deflection driving <torque> is proportional to $NI=N\mathcal E/(R_s+aN^2)$. Its derivative has the sign of $R_s-aN^2$, so the maximum occurs at $R_{\rm coil}=R_s$. Geometry, damping and finite response time can alter the practical optimization; this is a conditional winding model, not a universal rule for all <galvanometers>.
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